Fusion systems realizing certain Todd modules

TL;DR

This paper investigates specific simple fusion systems over finite 3-groups involving Todd modules of Mathieu groups and demonstrates their isomorphism to 3-fusion systems of almost simple groups, providing new characterizations of Conway's sporadic groups.

Contribution

It introduces a family of fusion systems involving Todd modules and proves their isomorphism to known 3-fusion systems, offering new local characterizations of sporadic groups.

Findings

Fusion systems are isomorphic to those of almost simple groups.

New 3-local characterizations of Conway's sporadic groups.

Identification of fusion systems involving Todd modules.

Abstract

We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the $3$-fusion systems of almost simple groups. As one consequence, we give new $3$-local characterizations of Conway's sporadic simple groups.